Abstract
Neighborhood graph based nonlinear dimensionality reduction algorithms, such as Isomap and LLE, perform well under an assumption that the neighborhood graph is connected. However, for datasets consisting of multiple clusters or lying on multiple manifolds, the neighborhood graphs are often disconnected, or in other words, have multiple connected components. Neighborhood graph based dimensionality reduction techniques cannot recognize both the local and global properties of such datasets. In this paper, a new method, called enhanced neighborhood graph, is proposed to solve the problem. The concept is to add edges to the neighborhood graph adaptively and iteratively until it becomes connected. Nonlinear dimensionality reduction can then be performed based on the enhanced neighborhood graph. As a result, both local and global properties of the data can be exactly recognized. In this study, thorough simulations on synthetic datasets and natural datasets are conducted. The experimental results corroborate that the proposed method provides significant improvements on dimensionality reduction for data with disconnected neighborhood graph.
Similar content being viewed by others
References
Lee JA, Verleysen M (2007) Nonlinear dimensionality reduction. Springer, Berlin
Verleysen M, Lee JA (2013) Nonlinear dimensionality reduction for visualization. In: 20th international conference neural information processing, ICONIP 2013. Springer, Berlin, pp 617–622
Jolliffe I (2005) Principal component analysis. Encyclopedia of statistics in behavioral science. Wiley, Hoboken
Borg I, Groenen PJF (2005) Modern multidimensional scaling: theory and applications. Springer, Berlin
DeMers D, Cottrell GW (1993) Non-linear dimensionality reduction. In: Advances in neural information processing systems 5, [NIPS conference]. Morgan Kaufmann Publishers Inc., pp 580–587
Van der Maaten LJP, Postma EO, Van den Herik HJ (2009) Dimensionality reduction: a comparative review. J Mach Learn Res 10:66–71
Hoffmann H (2007) Kernel PCA for novelty detection. Pattern Recognit 40:863–874
Hinton GE, Salakhutdinov RR (2006) Reducing the dimensionality of data with neural networks. Science 313:504–507
Mohebi E, Bagirov A (2016) Constrained self organizing maps for data clusters visualization. Neural Process Lett 43:849–869
Lee JA, Peluffo-Ordóñez DH, Verleysen M (2015) Multi-scale similarities in stochastic neighbour embedding: reducing dimensionality while preserving both local and global structure. Neurocomputing 169:246–261
Yang J, Fan L (2014) A novel indefinite kernel dimensionality reduction algorithm: weighted generalized indefinite kernel discriminant analysis. Neural Process Lett 40:301–313
Sammon JW (1969) A nonlinear mapping for data structure analysis. IEEE Trans Comput 18:401–409
Demartines P, Herault J (1997) Curvilinear component analysis: a self-organizing neural network for nonlinear mapping of data sets. IEEE Trans Neural Netw 8:148–154
Wan M, Lai Z, Jin Z (2011) Locally minimizing embedding and globally maximizing variance: unsupervised linear difference projection for dimensionality reduction. Neural Process Lett 33:267–282
Wang F, Zhang D (2013) A new locality-preserving canonical correlation analysis algorithm for multi-view dimensionality reduction. Neural Process Lett 37:135–146
Zhou Y, Sun S (2016) Local tangent space discriminant analysis. Neural Process Lett 43:727–744
Roweis ST, Saul LK (2000) Nonlinear dimensionality reduction by locally linear embedding. Science 290:2323–2326
Tenenbaum JB, De Silva V, Langford JC (2000) A global geometric framework for nonlinear dimensionality reduction. Science 290:2319–2323
Belkin M, Niyogi P (2003) Laplacian eigenmaps for dimensionality reduction and data representation. Neural Comput 15:1373–1396
Donoho DL, Grimes C (2003) Hessian eigenmaps: locally linear embedding techniques for high-dimensional data. Proc Natl Acad Sci 100:5591–5596
Saul LK, Roweis ST (2003) Think globally, fit locally: unsupervised learning of low dimensional manifolds. J Mach Learn Res 4:119–155
Zhang Z, Zha H (2004) Principal manifolds and nonlinear dimensionality reduction via tangent space alignment. J Shanghai Univ 8:406–424
Coifman RR, Lafon S (2006) Diffusion maps. Appl Comput Harmon Anal 21:5–30
Lee JA, Verleysen M (2005) Nonlinear dimensionality reduction of data manifolds with essential loops. Neurocomputing 67:29–53
Weinberger KQ , Sha F, Saul LK (2004) Learning a kernel matrix for nonlinear dimensionality reduction. In: Proceedings of the twenty-first international conference on Machine learning. ACM, Banff, p 106
Mekuz N, Tsotsos J (2006) Parameterless Isomap with adaptive neighborhood selection. Pattern Recognit. Springer, Berlin, pp 364–373
Samko O, Marshall AD, Rosin PL (2006) Selection of the optimal parameter value for the Isomap algorithm. Pattern Recognit Lett 27:968–979
Zhang Z, Wang J, Zha H (2012) Adaptive manifold learning. IEEE Trans Pattern Anal Mach Intell 34:253–265
Jia W et al (2008) Adaptive neighborhood selection for manifold learning. In: International conference on machine learning and cybernetics, 2008
Song Y et al (2008) A unified framework for semi-supervised dimensionality reduction. Pattern Recognit 41:2789–2799
de Ridder D et al (2003) Supervised locally linear embedding. Artificial neural networks and neural information processing—ICANN/ICONIP 2003. Springer, Berlin, pp 333–341
Huang Y, Xu D, Nie F (2012) Semi-supervised dimension reduction using trace ratio criterion. IEEE Trans Neural Netw Learn Syst 23:519–526
Zhang Z, Chow TWS, Zhao M (2013) M-Isomap: orthogonal constrained marginal Isomap for nonlinear dimensionality reduction. IEEE Trans Cybern 43:180–191
Nene SA, Nayar SK, Murase H (1996) Columbia object image library (COIL-20). Columbia University, New York
Liu X, Lu H, Li W (2010) Multi-manifold modeling for head pose estimation. In: 2010 IEEE international conference on image processing
Valencia-Aguirre J et al (2011) Multiple manifold learning by nonlinear dimensionality reduction. Springer, Iberoamerican Congress on Pattern Recognition
Torki M , Elgammal A, Lee CS (2010) Learning a joint manifold representation from multiple data sets. In: 2010 20th international conference on pattern recognition (ICPR). IEEE
Hadid A, Pietikäinen M (2003) Efficient locally linear embeddings of imperfect manifolds. In: Machine learning and data mining in pattern recognition: third international conference, MLDM 2003 Proceedings. Springer, Berlin, pp 188–201
Lee C-S, Elgammal A, Torki M (2016) Learning representations from multiple manifolds. Pattern Recognit 50:74–87
Yan S et al (2007) Graph embedding and extensions: a general framework for dimensionality reduction. IEEE Trans Pattern Anal Mach Intell 29:40–51
Lee CY (1961) An algorithm for path connections and its applications. IRE Trans Electron Comput EC–10:346–365
Tarjan R (1972) Depth-first search and linear graph algorithms. SIAM J Comput 1:146–160
Hopcroft J, Tarjan R (1973) Algorithm 447: efficient algorithms for graph manipulation. Commun ACM 16:372–378
Weyrauch B et al (2004) Component-based face recognition with 3D morphable models. In: Proceedings of the 2004 conference on computer vision and pattern recognition workshop (CVPRW’04), vol 05. IEEE Computer Society, p 85
Lee JA et al (2013) Type 1 and 2 mixtures of Kullback–Leibler divergences as cost functions in dimensionality reduction based on similarity preservation. Neurocomputing 112:92–108
Lee JA, Verleysen M (2014) Two key properties of dimensionality reduction methods. In: 2014 IEEE symposium on computational intelligence and data mining (CIDM). IEEE
Acknowledgements
This work is partially supported by the National Science Foundation of China under Grant no.61601112, the Fundamental Research Funds for the Central Universities and DHU Distinguished Young Professor Program. This work is also partially supported by the Natural Science Foundation of China under grant no. 61572156.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Fan, J., Chow, T.W.S., Zhao, M. et al. Nonlinear Dimensionality Reduction for Data with Disconnected Neighborhood Graph. Neural Process Lett 47, 697–716 (2018). https://doi.org/10.1007/s11063-017-9676-5
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11063-017-9676-5